How to fit a smooth curve on discrete data.

Hello all, I want to produce an equation that can develop a continous smooth curve (does not matter whether it follow any distribution or any plot) which connect the data given below. Using that equation I can interpolate data in between but I want a smoooth curve not a discrete curve. Can anyone please help me with this.

x = [3, 2.5, 2, 1.5, 1, 0.5]

y = [1.8, 1.75, 1.71, 1.55, 0.8, 1.25]

plot(x,y)

Answer 1

[Torsten]

Maybe something like this:

x = [3, 2.5, 2, 1.5, 1, 0.5];
y = [1.8, 1.75, 1.71, 1.55, 0.8, 1.25];
plot(x,y)
xx = linspace(x(1),x(end),100);
yy = interp1(x,y,xx,'cubic');
hold on
plot(xx,yy)
hold off

[Raj Arora]

Thanks Torsten, yes I was looking for the similar thing.

[Alex Sha]

If want a clear function, refer to below:

y = p1/(p2+(x-p3)^2)+p4;

Parameter	Best Estimate     
---------	-------------     
p1       	-0.141207212128963
p2       	0.119890467438701 
p3       	0.860890438017812 
p4       	1.81431810550217

[Raj Arora]

Alex can you tell me the procedure how you come up with this equaiton and values of p1 p2 p3 p4?

[Torsten]

Alex's idea is to approximate your data by a curve. This curve does not pass exactly through each of your data points, but gives a good approximation over the complete interval with a small number of coefficients.

x = [3, 2.5, 2, 1.5, 1, 0.5];
y = [1.8, 1.75, 1.71, 1.55, 0.8, 1.25];
fun = @(p,x) p(1)./(p(2)+(x-p(3)).^2)+p(4) ;
fun1 = @(p) fun(p,x)-y;
p0 = [-0.1;0.1;0.9;2] ;
sol = lsqnonlin(fun1,p0)

Local minimum possible.

lsqnonlin stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

sol = 4×1

-0.1412 0.1199 0.8609 1.8143

hold on
plot(x,y,'o')
xx = x(1):-0.01:x(end);
plot(xx,fun(sol,xx))
hold off